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In one dimension, the slope of a function, f(x), is described by a single number, df/dx.

In higher dimensions, the slope depends on the direction. For example, if f=x+2y, moving one unit x-ward increases f by 1 so the slope in the x direction is 1, but moving one unit y-ward increases f by 2 so the slope in the y direction is 2.

It turns out that we can describe the slope in n dimensions with just n numbers, the partial derivatives of f.

To calculate them, we differentiate with respect to one coordinate, while holding all the others constant. They are written using a ∂ rather than d. E.g.